Narrow Results

In this work, we study the Hénon–Heiles Hamiltonian, as a paradigm of open Hamiltonian systems, in the presence of different kinds of perturbations as dissipation, noise and periodic forcing, which are very typical in different physical situations. We focus our work on both the effects of these perturbations on the escaping dynamics and on the basins associated to the phase space and to the physical space. We have also found, in presence of a periodic forcing, an exponential-like decay law for the survival probability of the particles in the scattering region where the frequency of the forcing plays a crucial role. In the bounded regions, the use of the OFLI2 chaos indicator has allowed us to characterize the orbits. We have compared these results with the previous ones obtained for the dissipative and noisy case. Finally, we expect this work to be useful for a better understanding of the escapes in open Hamiltonian systems in the presence of different kinds of perturbations.

We consider the theory and applications of bivariate fractal interpolation surfaces constructed as attractors of iterated function systems. Specifically, such kind of surfaces constructed on rectangular domains have been used to demonstrate their efficiency in computer graphics and image processing. The methodology followed is based on the labeling used for the vertices of the rectangular domain rather than on the constraints satisfied by the contractivity factors or the boundary data.

This paper gives an up-to-date account of chaos and fractals, in a popular pictorial style for the general scientific reader. A brief historical account covers the development of the subject from Newton’s laws of motion to the astronomy of Poincaré and the weather forecasting of Lorenz. Emphasis is given to the important underlying concepts, embracing the fractal properties of coastlines and the logistics of population dynamics. A wide variety of applications include: NASA’s discovery and use of zero-fuel chaotic “superhighways” between the planets; erratic chaotic solutions generated by Euler’s method in mathematics; atomic force microscopy; spontaneous pattern formation in chemical and biological systems; impact mechanics in offshore engineering and the chatter of cutting tools; controlling chaotic heartbeats. Reference is made to a number of interactive simulations and movies accessible on the web.

In this paper, the membership function in fuzzy systems is used in the Diffusion Limited Aggregation (DLA) model to investigate the fractal diffusion of soot particles from diesel engine emissions. The transformation of the morphology of soot particle aggregates and the control of fractal diffusion of soot particles are investigated by analyzing the nonlinear relationship between the motion steps and angles of diffusing particles. The simulation results demonstrate that the morphology of the aggregates varies from loose to compact by changing the particles’ motion steps and angles in membership functions. Meanwhile, the Ballistic Aggregation (BA)-like aggregates are obtained. Furthermore, the control of the morphology of soot particle aggregates is realized, which makes the settlement of the aggregates become easier. This will provide a reference for further understanding the growth mechanism of soot particle diffusion and enhancing the purification technology of the soot particles.

It is not common in applied sciences to realize simulations which depict fractal representation in attractors’ dynamics, the reason being a combination of many factors including the nature of the phenomenon that is described and the type of differential operator used in the system. In this work, we use the fractal-fractional derivative with a fractional order to analyze the modified proto-Lorenz system that is usually characterized by chaotic attractors with many scrolls. The fractal-fractional operator used in this paper is a combination of fractal process and fractional differentiation, which is a relatively new concept with most of the properties and features still to be known. We start by summarizing the basic notions related to the fractal-fractional operator. After that, we enumerate the main points related to the establishment of proto-Lorenz system’s equations, leading to the $n$th cover of the proto-Lorenz system that contains $n$ scrolls ($n\in \mathbb{N}$). The triple and quadric cover of the resulting fractal and fractional proto-Lorenz system are solved using the Haar wavelet methods and numerical simulations are performed. Due to the impact of the fractal-fractional operator, the system is able to maintain its chaotic state of attractor with many scrolls. Additionally, such attractor can self-replicate in a fractal process as the derivative order changes. This result reveals another great feature of the fractal-fractional derivative with fractional order.

In this paper, we investigate coexisting strange nonchaotic attractors (SNAs) in a quasiperiodically forced system. We also describe the basins of attraction for coexisting attractors and identify the mechanism for the creation of coexisting attractors. We find three types of routes to coexisting SNAs, including intermittent route, Heagy–Hammel route and fractalization route. The mechanisms for the creation of coexisting SNAs are investigated by the interruption of coexisting torus-doubling bifurcations. We characterize SNAs by the largest Lyapunov exponents, phase sensitivity exponents and power spectrum. Besides, the SNAs with extremely fractal basins exhibit sensitive dependence on the initial condition for some particular parameters.

We examine the orbital dynamics in a new Hénon–Heiles system with an additional gravitational potential, by classifying sets of starting conditions of trajectories. Specifically, we obtain the results on how the total orbital energy along with the transition parameter influence the overall dynamics of the massless test particle, as well as the respective time of escape/collision. By using modern diagrams with color codes we manage to present the different types of basins of the system. We show that the character of the orbits is highly dependent on the energy and the transition parameter.

During the last few years, many algorithms have been proposed in particular for face recognition using classical 2-D images. However, it is necessary to deal with occlusions when the subject is wearing sunglasses, scarves and such. In the same way, ear recognition is arising as a new promising biometric for people recognition, even if the related literature appears to be somewhat underdeveloped. In this paper, several hybrid face/ear recognition systems are investigated. The system is based on IFS (Iterated Function Systems) theory that are applied on both face and ear resulting in a bimodal architecture. One advantage is that the information used for the indexing and recognition task of face/ear can be made local, and this makes the method more robust to possible occlusions. The distribution of similarities in the input images is exploited as a signature for the identity of the subject. The amount of information provided by each component of the face and the ear image has been assessed, first independently and then jointly. At last, results underline that the system significantly outperforms the existing approaches in the state of the art.

We introduce the k-peg Hanoi automorphisms and Hanoi self-similar groups, a generalization of the Hanoi Towers groups, and give conditions for them to be contracting. We analyze the limit spaces of a particular family of contracting Hanoi groups, , and show that these are the unique maximal contracting Hanoi groups under a suitable symmetry condition. Finally, we provide partial results on the contraction of Hanoi groups with weaker symmetry.

We explore the relationship between the limit spaces of contracting self-similar groups and self-similar structures. We give the condition on a contracting group such that its limit space admits a self-similar structure, and also the condition such that this self-similar structure is post-critically finite (p.c.f.). We then give necessary and sufficient conditions for a p.c.f. self-similar structure to be the limit space of a contracting self-similar group. When these conditions hold we give a construction of the contracting group. Finally, we illustrate our results with several examples.

There are three constructions of which I know that yield higher dimensional analogues of Sierpinski’s triangle. The most obvious is to remove the open convex hull of the midpoints of the edges of the $n$-simplex. The complement is a union of simplices. Continue the removal recursively in each of the remaining sub-simplices. The result is an uncountably infinite figure in $n$-dimensional space that is Cantor-like in a manner analogous to the Sierpinski triangle. A countable analogue is obtained by means of playing the chaos game in the $n$-simplex. In this “game” one chooses a random $(n+1)$-ary sequence; starting from the initial point (that is identified with a vertex of the simplex), one continues to plot points by moving half-again as much towards the next point in the sequence. The resulting plot converges to the figure described above. Similarly, coloring the multinomial coefficients black or white according to their parity results in a similar figure, when the $n$-dimensional analogue of the Pascal triangle is rescaled and embedded in space.

In this paper we first provide the open chain and the closed chain method to calculate the partition functions of the typical fractal lattices, i.e. a special kind of Sierpinski carpets(SC) and the triangular Sierpinski gaskets(SG). We then apply knot theory to fractal lattices by changing lattice graphs into link diagrams according to the interaction models, and explicitly obtain the partition functions of a special SC for the edge interaction models. These partition functions are also the knot invariants of the corresponding link diagrams. This is the first time that topology enters into fractals.

In the present paper, a modeling in the complex space is combined with complex-valued fractional Brownian motion to get some new results in biological systems. The rational of this approach is as follows. Biological dynamics which evolve continuously in time but are not time differentiable, necessarily exhibit random properties. These random features appear also as a result of the randomness of the proper time of biological systems. Usually, this is taken into account by using white noises that is to say fractals of order two. Fractals of order n larger than two are more suitable for increments with large amplitudes, and they may be introduced by using either real-valued fractal noises with long range memory or Brownian motions with independent increments, which are necessarily complex-valued. In the later case, we are then led to describe biological systems in the complex plane. After some background on the complex-valued fractional Brownian motion, we shall deal successively with population growth, information thermodynamics of order n, nonequilibrium phase transition via fractal noises and complexity of Markovian processes via the concept of informational divergence.

We present a dynamical model for the particle-cluster aggregation process of marine snow. The model consists of only two parts, i.e. single cells and one type of aggregates. The parameters are settling and decomposition of aggregates; growth and aggregation of single cells. There are developed numerical results of the system.

In biochemical dynamical systems during each transition between periodical behaviors, all metabolic intermediaries of the system oscillate with the same frequency but with different phase-shifts. We have studied the behavior of phase-shift records obtained from random transitions between periodic solutions of a biochemical dynamical system. The phase-shift data were analyzed by means of Hurst's rescaled range method (introduced by Mandelbrot and Wallis). The results show the existence of persistent behavior: each value of the phase-shift depends not only on the recent transitions, but also on previous ones. In this paper, the different kind of periodic solutions were determined by different small values of the control parameter. It was assessed the significance of this results through extensive Monte Carlo simulations as well as quantifying the long-range correlations. We have also applied this type of analysis on cardiac rhythms, showing a clear persistent behavior. The relationship of the results with the cellular persistence phenomena conditioned by the past, widely evidenced in experimental observations, is discussed.

We used a multifractal approach to characterize scale by scale, the remotely sensed visible and thermal-infrared volcanic field, at Kilauea Volcano, Hawaii, USA. Our results show that (1) the observed fields exhibit a scaling behavior over a resolution range of ~ 2.5 m to 6 km, (2) they show a strong multifractality, (3) the multifractal parameters α, C₁ and H are sensitive to volcanic structural classes such as vent cones, lava ponds and active to inactive lava flows, (4) vegetation area and volcanic gas plumes have a strong effect on the multifractal estimates, and (5) vegetation and cloud-free images show statistical characteristics due to topography related albedo in the visible and predominantly solar heating in the thermal infrared wavelengths.

Combat data collected from World War II and a cellular automaton combat model called MANA are shown to display fractal properties. This strongly supports our earlier hypotheses as to the nature of combat attrition data. It also provides a method by which we can judge a combat model's ability to produce realistic synthetic combat data. The data appear to display properties extremely similar to those of the fractal cascade models used to describe turbulent dynamics. Interestingly, the fractal parameters appear to depend on how the model is set up, implying that they are determined by the boundary and initial conditions. Examination of the dynamical rules used in the MANA model simulation suggests that the model entities need to respond to changes in the level of order on the battlefield grid for fractal behavior to occur. Such data imply that the entropy of the battlefield is dependent on the scale at which it is examined. We speculate that such formations in a military case effectively act to isolate the highest level of command from disorder in the lowest. If disorder within a force grows to the point where that force can no longer maintain a fractal-like distribution, the force distribution may tend to become uniformly random, effectively destroying its viability as a combat unit.

Usage of a deterministic fractal-multifractal (FM) procedure to model high-resolution rainfall time series, as derived distributions of multifractal measures via fractal interpolating functions, is reported. Four rainfall storm events having distinct geometries, one gathered in Boston and three others observed in Iowa City, are analyzed. Results show that the FM approach captures the main characteristics of these events, as the fitted storms preserve the records' general trends, their autocorrelations and spectra, and their multifractal character.

The cloud radiances and atmospheric dynamics are strongly nonlinearly coupled, the observed scaling of the former from 1 km to planetary scales is prima facae evidence for scale invariant dynamics. In contrast, the scaling properties of radiances at scales <1 km have not been well studied (contradictory claims have been made) and if a characteristic vertical cloud thickness existed, it could break the scaling of the horizontal radiances. In order to settle this issue, we use ground-based photography to study the cloud radiance field through the range scales where breaks in scaling have been reported (30 m to 500 m). Over the entire range 1 m to 1 km the two-dimensional (2D) energy spectrum (E(k)) of 38 clouds was found to accurately follow the scaling form E(k)≈ k^-β where k is a wave number and β is the spectral exponent. This indirectly shows that there is no characteristic vertical cloud thickness, and that "radiative smoothing" of cloud structures occurs at all scales. We also quantitatively characterize the type of (multifractal) scaling showing that the main difference between transmitted and reflected radiance fields is the (scale-by-scale) non-conservation parameter H. These findings lend support to the unified scaling model of the atmosphere which postulates a single anisotropic scaling regime from planetary down to dissipation scales.

The study of basic fractal geometry can help build students' enthusiasm for learning early in their undergraduate careers. To most undergraduate students, fractals are new, visually appealing, useful, and mathematically accessible. As a result, fractals can be an effective vehicle for introducing and reinforcing multiple modes of learning, which at many institutions is one of the main goals of general first-year undergraduate education. This article describes how fractals are used in one institution's "Freshman Seminar" program to help accomplish these goals.